Peaceman–Rachford splitting for a class of nonconvex optimization problems

Abstract

We study the applicability of the Peaceman–Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.

Keywords

G. Li was partially supported by a Research Grant from Australian Research Council. T. Liu was supported partly by the AMSS-PolyU Joint Research Institute Postdoctoral Scheme. T. K. Pong was supported partly by Hong Kong Research Grants Council PolyU253008/15p.

Appendix: concrete numerical examples

In this appendix, we provide some simple and concrete examples illustrating the different behaviors of the classical PR splitting method, the classical DR splitting method and our proposed PR splitting method (33).

The first example shows that, even in the convex setting, the classical PR splitting method can be faster than the classical DR splitting method, and our proposed PR method can outperform the classical DR method for some particular choice of the parameter \(\gamma \). The second example on nonconvex feasibility problem shows that the classical PR method can diverge while our proposed PR method converges linearly to a solution for the feasibility problem.

Thus, for \(\gamma =0.01\), our proposed PR method (33) is faster than the classical DR method for this example.

Example 2

(Classical PR method vs. the proposed PR method) Let \(C=\{(0,0)\}\) and \(D=\big (\{0\} \times \mathrm{I}\!\mathrm{R}\big ) \cup \big (\mathrm{I}\!\mathrm{R}\times \{0\}\big )\). We consider the feasibility problem of finding a point in the intersection of C and D. We start with the initial point \(x^0=(a,0)\) with \(a\ne 0\). Then, the classical PR splitting method applies (6) to \(f(x)=\delta _C(x)\) and \(g(x)=\delta _D(x)\) for all \(x \in \mathrm{I}\!\mathrm{R}^2\), and reduces to

where the formula for the z-update follows from the fact that \(x^t\), \(y^t \in \mathrm{I}\!\mathrm{R}\times \{0\}\subset D\), and so is \(2y^{t+1}-x^t\) by the construction. Hence, the proposed PR method (51) converges to \((0,0) \in C \cap D\) linearly in this case.